Submersions from compact flat manifoldThe point is: if $H^a(\tilde{N})$ is the highest nonzero cohomology group of $\tilde{N}$, and $H^b(F')$ the highest cohomology group of $F'$, then in the second page of the spectral sequence the term $E_2^{a,b}$ is nonzero, and no nonzero differentials land on it, or leave it. For this fact to be true it is important that both cohomologies of $F'$ and $\tilde{N}$ have only a finite number of nonzero cohomology group, so this passage does not apply in the situations you described. It follows that $E^{a,b}_2$ survives to the infinity page, and in particular $H^{a+b}(\tilde{M})\neq 0$.

Feb1

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Submersions from compact flat manifoldIgor: As you say, you precompose the pullback with the universal cover. By the exact sequence in homotopy the fiber F′ is connected. What is not working in the second paragraph?

Dense subgroups of Lie Groups@Misha: thanks for the explanation. One possible concept of complexity is the growth of H. I am trying to understand what kind of compact manifolds can admit H as fundamental group (or a quotient of it).

(Non)-exoticness of a diffeomorphism of a sphereHi Ryan, thanks for the answer first of all. By torus i really meant $(S^1)^n$, even though I didn't say how that came up. I thought it was going to make the question heavier, and all I really wanted was to get to question 3.

Euler characteristic and universal coverHi Johannes! actually, my problem is that $\tilde{M}$ retracts to a lie group, but is not one. in particular, i don't see how i can make $\pi(M)$ act on the retraction, in a free fashion